They have been through a lot together, Andre Agassi and Wimbledon. Their special relationship has survived standoffishness, mutual mistrust and a lot of very bad-hair days and it has emerged the stronger for it. Wimbledon never quite seemed likely to embrace the Lycra-loving chancer from Las Vegas, who arrived in a blaze of hype, but 16 years on he can expect the intense emotional welcome reserved for the greatest emotional favourites of the past, Jaroslav Drobny and Rod Laver, John McEnroe and Boris Becker.

Agassi never dreamed he would fall in love with Wimbledon. But then he never thought either that he would be approaching the tournament in his thirty-fourth year as the number-one player in the world (the oldest man to have held that distinction). He never thought he would have a great chance to become the first father to win the singles since Pat Cash in 1987. And, perhaps most of all, he never thought he would be the last man standing here among his peers.

Anyone who followed the first half of Agassi's career would, likewise, have laid strong odds against the fact that out of that golden generation of American players, Sampras and Agassi, Chang and Courier, who turned professional when Ronald Reagan was still president, it would be him, the eldest and least dogged, who, a thousand and one matches later, is still playing. The indefatigable Michael Chang has long since run himself into the ground; the iron-willed Jim Courier found his desire was lacking three years ago; and even the great Pete Sampras, after what looks increasingly like a last hurrah at the 2002 US Open, cannot get himself excited about a farewell crack at Wimbledon. Agassi, though, arrives here, having won four of the first five tournaments of the year, including the Australian Open, with eyes shining.

It was not always so, of course. In his first five years on the tour Agassi played in SW19 for a grand total of 67 minutes, the length of time it took Henri Leconte to beat him on Court 2 in 1987. In that hour or so he convinced himself of three things: 'I had no wish to be on the grass. I didn't feel it was tennis. It was inconvenient in my schedule.'

For four years he did not make the trip at all. Then, he says, he supposes he began to grow up. He came to an understanding of 'the place Wimbledon has in this great sport of tennis', and he began to find that to miss it 'was my loss'.

The example of Bjorn Borg had given him faint hope that you could succeed here from the back of the court, and he proved it to himself when he returned in 1991 and reached the quarter-finals. The following year he came back to win his long overdue first grand-slam title. Asked at Queen's if he remembered much of that final, he smiled to himself and said: 'No, not very much,' before reeling off a shot-by-shot account of his epic duel with Goran Ivanisevic.

'Two all, two break points against me, serve out wide, forehand approach, his chipped lob, my overhead?' And so on, until he had to laugh. That match, in which Ivanisevic served more than a set-and-a-half of aces and still lost, was Agassi's coming of age.

'When I look at it in photographs now,' he says, 'it is for me about so much more than tennis. Cradling the trophy was me embracing what was not even a dream, but more fears and doubts and questions and losing three grand-slam finals, and being told by more than just the media that this was something I couldn't win. That is the kind of moment that makes sport great: it had so much more to it for me than the game of tennis.'

The man who learned his game in the desert heat, a racket taped to his hand at the age of two, praying for rain at five years old so he would not have to practise, still does not believe, however, that professional tennis should be played on a lawn. 'If it wasn't for Wimbledon, then no, there's no place for it. I think that Wimbledon makes the grass more than the grass makes Wimbledon. Grass highlights the shot-making side of the game, but at the expense of the whole strategic side of things. You get excited if you see five balls go back and forth. But still,' he concedes, 'to be out there [on Centre Court] is the greatest feeling in the world.'

As Agassi nears the end of his career, he is, it seems, more than ever prepared to appreciate that feeling. When I interviewed him even three years ago, he was far more tortured about the demands of the sport, still a little in thrall to the pressures of being a prodigy. Talking about whether to retire, he could sound a lot like Hamlet interrogating his mortality.

'At the end of the day I need to do it or not do it,' he said. 'That has always been the question for me. The question of not doing this thing I was kind of born to do has been with me for 15 years. It's been hard to ignore it at times. But I've had to think about it and answer it, get on the plane and do it again...' Now, you have the sense that he has fewer doubts, and that whereas in the past he felt an obligation to do it, now he has a need. 'The desire's all there,' Agassi said last week.

'When you have played a thousand matches you have seen it all. You know how things unfold. And you know what you have to do to change that to make things better for yourself. I had that experience this year in Australia.' Agassi played almost the perfect tournament, dropping only one set in the fortnight and winning the final in under an hour-and-a-half against the German Rainer Schuettler. 'I would stay on another five years to have one more day like that.'

Agassi would love more than anything for that day to be in a fortnight's time. But he long ago learned that to win at Wimbledon, you have to be prepared to wait. The greatest match he played here, he believes, was probably his semi-final win over McEnroe in 1992. 'That was a clean match start to finish,' he says, and for Agassi cleanliness is always next to godliness. Some of the wisdom that allowed him to succeed here came from McEnroe, who bowed out after that semi-final defeat, and gave Agassi the benefit of his experience about facing down the biggest servers. 'He told me not to expect to have a lot of chances, but to concentrate on the few opportunities I did get and work out what to do with them. Goran served 38 aces in that final [1992] and I spent a lot of time just walking back and forth from this court to that court. But I didn't let it bother me.' He just waited.

He is more than canny enough to also know that, having waited 11 years since his last triumph, this year he has as good a chance to repeat that performance as he has ever had. The great serve and volleyers of his era have all come and gone: Edberg, Becker, Ivanisevic, Rafter and most particularly Sampras.

The only real grass-court natural who remains, Tim Henman, is someone he respects ('He's the real deal, out there trying to win the tournament') but hardly fears. Of the younger pretenders, only Lleyton Hewitt can claim comparable mental strength, a little bit of which was undermined in last year's US Open semi-final when Agassi dominated. Andy Roddick may have beaten him at Queen's but it is much harder to imagine him doing the same on the slower courts in the second week at Wimbledon. With all this in mind, Agassi has come more than usually ready.

One of the reasons for his latest resurgence over the last year or so is, he believes, his new coach Darren Cahill (once Hewitt's mentor), who took over from Brad Gilbert in 2001.

'Brad had done incredible things with my tennis over the years and taught me how to think for myself out there,' Agassi says. 'But Darren has come along and made me better. I need to get better to stay the same and I feel like I'm still improving at 33.'

He is as fit as he has ever been. He keeps his trophies these days in his gym - 'my working environment' - as a reminder of 'what it is we have done and what it is we strive for'. (He's not sure where his wife Steffi Graf keeps all hers, though 'She could lose half and still not have enough shelf space.')

He has also made the commitment of coming here for nearly a full month. In the past he has rarely played on grass before Wimbledon, believing that it has a bad effect on the rest of the season 'because you start getting little kinks in your rhythm and your swing doesn't stay clean'. This year, though, notably, he has not only played Queen's but also visited the All England Club with his coach at the start of the summer to imagine the possibilities. 'I said to Darren, you're the guest, I'm the member, so don't make me look bad around here.'

All of this effort is directed at the one compulsion Agassi retains: that he should not just be along for the ride. 'For me, doing this a long time is one thing, but I want to be out here with a chance to win. I take a lot of pride in the fact that if I play my tennis on any given day I can still win. That's a necessary feeling for me these days.'

He is more than lucky, he says, to have a wife who understands exactly this necessity. If it was a choice either to stay at home with his family or go on tour, one or the other, then he says it would be no choice at all. But Steffi never gives him that choice.

'One of the advantages of having a partner who has herself won 22 grand slams is that she knows exactly the sacrifices that are required,' he says. 'She knows what it means to go out there when your body's not right or your confidence is low, and she knows when things don't need to be said, and you appreciate all of those moments.'

Graf had made a bet with her husband that if he won the Australian title she would team up with him for mixed doubles in Paris. As it turned out a second pregnancy prevented her from keeping her side of the bargain: 'It's amazing,' Agassi says, grinning, 'the lengths she'll go to to not to play with me.' And he laughs when he says that he's fearing the worst.

'They say when you go from one to two children it's not doubling your work, it's increasing it tenfold.' With this in mind he says he has no idea how long he will carry on for, but the greatest returner of a tennis ball the game has seen is more than happy to take what is thrown at him. 'I'm not going to plan out a farewell year,' he says.

'The journey is what it is about for me, the experiences that come along, the end will have its own say.' You have the sense that it won't be speaking up for a couple of years yet.

· Tim Adams' new book Being John McEnroe is available from The Observer Books Service. To order a copy for £8 plus p&p (rrp £10), call 0870 066 7989. Published by Yellow Jersey Press.

All about Agassi

· When he was three he was labelled a tennis prodigy and by the age of four was practising with Björn Borg, Ilie Nastase and Harold Solomon

· Agassi's father, a former boxer who participated in the 1952 Olympics, introduced his son to tennis

· He married actress Brooke Shields in 1997 but he filed for divorce in 1999

· In 1986, he emerged from the Nick Bollettieri tennis academy as the latest all-American tennis talent

· He married Steffi Graf in October 2001 in Las Vegas. A week later she gave birth to their first child, Jaden Gil

· In June 1999 Agassi became only the fifth player to win all four grand slams

· He did not win a grand-slam title in 1996, but won an Olympic gold medal

· In 1994 Agassi became the first unseeded player since 1930 to win the US Open

· Agassi recaptured the world number one ranking after his quarter-final victory over Malisse at Queen's last week

· In 1992 he won more points from second serves than anyone else on the circuit

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